nuevos enfoques sobre medición y escalamientos de desplazamientos inmiscibles en medios porosos

Nuevos Enfoques sobre Medición y Escalamientos de Desplazamientos

Inmiscibles en Medios Porosos

Nuevos Enfoques sobre Medición y Escalamientos de Desplazamientos

Inmiscibles en Medios Porosos

Marcelo Crotti - INLAB S.A.Marcelo Crotti - INLAB S.A.

W W W DarcyW W W Darcyhat´shat´s rongrong ithith

A journey through the fantasy of Relative Permeability A journey through the fantasy of Relative Permeability ConceptConcept

UADE – Primer Encuentro Nacional sobre Simulación de ReservoriosUADE – Primer Encuentro Nacional sobre Simulación de Reservorios

ContentsContents

A brief analysis of the origin and uses of DarcyA brief analysis of the origin and uses of Darcy´s law in Reservoir Engineering.´s law in Reservoir Engineering.

Differences between: Differences between: Conduction of fluids.Conduction of fluids. Injection of fluids.Injection of fluids. Production of fluids.Production of fluids.

Limitations of the Relative Permeability Limitations of the Relative Permeability concept.concept.

Different Sceneries.Different Sceneries. Conclusions.Conclusions.


Darcy´s Law (I)Darcy´s Law (I)

Darcy´s law describes homogeneous fluid Darcy´s law describes homogeneous fluid flow through linear porous materials. This law, flow through linear porous materials. This law, which relates flow velocity with pressure which relates flow velocity with pressure gradient, has three components. gradient, has three components. A geometric factor given by length and area of A geometric factor given by length and area of

porous medium.porous medium. A fluid related factor (viscosity). A fluid related factor (viscosity). A property of the porous material (Permeability).A property of the porous material (Permeability).

Q = K . A . Q = K . A . P / (µ . L) P / (µ . L)


Darcy´s Law (II)Darcy´s Law (II)

Permeability is usually defined as Permeability is usually defined as

““The ability of a porous material to The ability of a porous material to conductconduct fluids".fluids".


Darcy´s Law (III)Darcy´s Law (III)

Only Only injectioninjection rate or rate or productionproduction rate can be rate can be measured experimentally.measured experimentally.

But... during the flow of an uncompressible But... during the flow of an uncompressible homogeneous fluid:homogeneous fluid: The rate of The rate of conductionconduction is equal to: is equal to: InjectionInjection rate and rate and ProductionProduction rate. rate.

So, measuring one of these rates is enough So, measuring one of these rates is enough to obtain the value of the other rates.to obtain the value of the other rates.


Darcy´s Law (IV)Darcy´s Law (IV)

For multiphase fluid flow, Darcy´s law was For multiphase fluid flow, Darcy´s law was “corrected” using a different factor for each “corrected” using a different factor for each flowing phase. This factor is known as flowing phase. This factor is known as relative permeability curve. relative permeability curve.

QQww = K . = K . KrKrww . A . . A . PPww / (µ / (µww . L) .......... . L) .......... [2][2]

QQoo = K . = K . KrKroo . A . . A . PPoo / (µ / (µoo . L) .......... . L) .......... [3][3]

The value of relative permeability is different The value of relative permeability is different for every fluid saturation. for every fluid saturation.


Darcy´s Law (V)Darcy´s Law (V)

However .....However ..... A new problem arises during unsteady multiphase A new problem arises during unsteady multiphase

fluid flow.fluid flow. The rate of The rate of conductionconduction of each phase of each phase losseslosses its its equivalenceequivalence

withwith InjectionInjection rate, and rate, and ProductionProduction rate. rate.

When more than one phase is flowing, it is not When more than one phase is flowing, it is not possible to measure possible to measure productionproduction rate or rate or injectioninjection rate rate to determine the to determine the conductionconduction rate of each phase. rate of each phase.

And Darcy´s law applied to each phase is based on And Darcy´s law applied to each phase is based on conductionconduction rate exclusively. rate exclusively.


Darcy´s Law (VI)Darcy´s Law (VI)

In order to solve this "inconsistency" two ways were In order to solve this "inconsistency" two ways were found. One of them during measurement, and the found. One of them during measurement, and the other during calculation.other during calculation. Experimentally a method was developed in order to re-Experimentally a method was developed in order to re-

create Darcy´s requirements (Injection = Conduction = create Darcy´s requirements (Injection = Conduction = Production). This methodology was known as “steady state”. Production). This methodology was known as “steady state”. Simultaneous Injection rate is maintained until production rate Simultaneous Injection rate is maintained until production rate

equals injection rate of each phase. equals injection rate of each phase. Through calculation, equations were solved during unsteady Through calculation, equations were solved during unsteady

state displacement in order to obtain values in a state displacement in order to obtain values in a dimensionless section. This is known as “unsteady state” or dimensionless section. This is known as “unsteady state” or Welge methodology. Welge methodology.


Darcy´s Law (VII)Darcy´s Law (VII)

Briefly, to allow the use of Darcy´s law during Briefly, to allow the use of Darcy´s law during multiphase fluid flow, it was necessary to generate a multiphase fluid flow, it was necessary to generate a unique saturation at the position of calculation. In this unique saturation at the position of calculation. In this way, once again: way, once again: InjectionInjection = = ConductionConduction = = ProductionProduction

During “steady-state” measurements the whole During “steady-state” measurements the whole sample has the same saturation.sample has the same saturation.

Using “unsteady-state” methodology, all calculations Using “unsteady-state” methodology, all calculations are made in a single point (the outlet face) were the are made in a single point (the outlet face) were the saturation is unique (point saturation).saturation is unique (point saturation).


A Question (I)A Question (I)

Both methodologies (“steady-state” and Both methodologies (“steady-state” and “unsteady-state”) gives the same result “unsteady-state”) gives the same result when applied to homogeneous media.when applied to homogeneous media.

However .....However ..... Is this a good answer for Reservoir Is this a good answer for Reservoir

Calculations?Calculations?


A Question (II)A Question (II)

In other words:In other words:The solution for the particular case whereThe solution for the particular case where

InjectionInjection = = ConductionConduction = = ProductionProduction

is useful foris useful forunsteady reservoir conditions?.unsteady reservoir conditions?.


Answer (I)Answer (I)

A very simple example will give a satisfactory answer for this A very simple example will give a satisfactory answer for this question.question.

Let us suppose that we have a thin, horizontal porous system.Let us suppose that we have a thin, horizontal porous system. If the system is filled with gas it is obvious that its ability to If the system is filled with gas it is obvious that its ability to

conductconduct water is null. Using Darcy law we can say that: water is null. Using Darcy law we can say that: Kw = 0 when Sw = 0Kw = 0 when Sw = 0

However.....However..... Although water cannot be “Although water cannot be “conductedconducted” there is no special ” there is no special

impediment to impediment to injectinject water in the porous system. An empty water in the porous system. An empty rock allows water rock allows water injectioninjection..

Although water can be Although water can be injectedinjected, water cannot be , water cannot be producedproduced until it reaches de outlet end.until it reaches de outlet end.


Answer (II)Answer (II)

Using this model we may ask:Using this model we may ask: Which is the system ability to conduct water whenWhich is the system ability to conduct water when

Sw = 50%?Sw = 50%? Half of the system with Sw = 100% and the other half with Half of the system with Sw = 100% and the other half with

Sw = 0%Sw = 0% There are two “well defined” capacities to conduct There are two “well defined” capacities to conduct

water at this moment: water at this moment: The capacity "X" (obtained through Darcy´s calculations) The capacity "X" (obtained through Darcy´s calculations)

where Sw = 100% where Sw = 100% A null capacity where Sw = 0A null capacity where Sw = 0

So.... Which is the overall value for the water So.... Which is the overall value for the water conduction ability?conduction ability?


Answer (III)Answer (III)

Option 1: Kw = X ? Option 1: Kw = X ? Option 2: Kw = 0 ? Option 2: Kw = 0 ? Option 3: Kw = X/2 ?Option 3: Kw = X/2 ? Option 4: ......Option 4: ...... While there is not a well defined While there is not a well defined ConductionConduction

ability, always exists a well defined ability, always exists a well defined InjectionInjection ability and also a well defined ability and also a well defined ProductionProduction ability. ability.


Real ScenariesReal Scenaries

In Reservoir Simulation through Darcy´s In Reservoir Simulation through Darcy´s calculations only calculations only ConductionConduction ability is used. ability is used.

For this reason Darcy´s equation is unable to For this reason Darcy´s equation is unable to reproduce reproduce InjectionInjection or or ProductionProduction rates in rates in unsteady systems.unsteady systems.

And .....And ..... All reservoirs are Unsteay Systems during All reservoirs are Unsteay Systems during

production.production.


Reference Frame in Natural Reservoirs (The actual world)

Reference Frame in Natural Reservoirs (The actual world)

In Reservoir Engineering frequently: In Reservoir Engineering frequently: Natural porous media are heterogeneous. Natural porous media are heterogeneous. Multiphase flow is the result of an equilibrium Multiphase flow is the result of an equilibrium

between viscous, capillary and gravity forces. This between viscous, capillary and gravity forces. This equilibrium varies with time and with physical equilibrium varies with time and with physical location. location.

Reservoir calculations are founded on average Reservoir calculations are founded on average phase saturation. phase saturation. In a single cell (Reservoir Simulation) In a single cell (Reservoir Simulation) The whole reservoir (Material Balance).The whole reservoir (Material Balance).

The properties of interest are The properties of interest are ProductionProduction and and InjectionInjection abilities. abilities.


Solutions (I)Solutions (I)

Every Scenery has its own solution.Every Scenery has its own solution. Typical situations are:Typical situations are:

Homogeneous systems with dominant viscous forces. Very Homogeneous systems with dominant viscous forces. Very few reservoirs fall in this category, but it is the typical few reservoirs fall in this category, but it is the typical laboratory scenery.. laboratory scenery..

Heterogeneous systems under viscous forces Heterogeneous systems under viscous forces predominance. Stratified reservoirs without connected predominance. Stratified reservoirs without connected layers. layers.

Heterogeneous systems under viscous and capillary forces Heterogeneous systems under viscous and capillary forces predominance. Stratified reservoirs with connected layers. A predominance. Stratified reservoirs with connected layers. A cross-flow happens as a consequence of imbibition cross-flow happens as a consequence of imbibition phenomena.phenomena.


Solutions (II)Solutions (II)

Other typical situations.Other typical situations. Heterogeneous systems with heavy oil where Heterogeneous systems with heavy oil where

imbibition phenomena could be dominant.imbibition phenomena could be dominant. Gravity dominated systems. Mainly in high Gravity dominated systems. Mainly in high

permeability rocks with remarkable thickness and permeability rocks with remarkable thickness and density differences and low viscosities (expanding density differences and low viscosities (expanding gas cap, basal aquifers, ...). gas cap, basal aquifers, ...).

Reservoirs dominated by capillary forces. Mainly Reservoirs dominated by capillary forces. Mainly in "tight sands“ reservoirs.in "tight sands“ reservoirs.


Conclusions (I)Conclusions (I)

Laboratory measurements are adequate to Laboratory measurements are adequate to describe the ability to describe the ability to conductconduct fluids in fluids in porous media with homogeneous fluids porous media with homogeneous fluids saturation.saturation.

In Reservoir Engineering, the ability to In Reservoir Engineering, the ability to injectinject or to or to produceproduce fluids in non-homogeneous fluids in non-homogeneous fluids saturation is needed. In unsteady fluids saturation is needed. In unsteady systems the ability to systems the ability to conductconduct fluids losses fluids losses its meaning and becomes useless.its meaning and becomes useless.


Conclusions (II)Conclusions (II)

In order to fill the gap, experimental measurements must be In order to fill the gap, experimental measurements must be made honoring real mechanisms occurring at reservoir scale.made honoring real mechanisms occurring at reservoir scale.

During scaling-up stage, the following points must be During scaling-up stage, the following points must be considered:considered: The dominant displacement mechanism.The dominant displacement mechanism. Laboratory “end point” measurements.Laboratory “end point” measurements. System geometry.System geometry. Heterogeneity.Heterogeneity.

The scaling-up process must be made by a multidisciplinary The scaling-up process must be made by a multidisciplinary team.team.

Relative permeability curves must be “constructed” for every Relative permeability curves must be “constructed” for every case. This operation must be based on available data and case. This operation must be based on available data and accepted reservoir models.accepted reservoir models.


W W W DarcyW W W Darcy

Thank you for your attentionThank you for your attention


ExamplesExamples

The Relationship between production and The Relationship between production and fluid saturation will be analyzed using two fluid saturation will be analyzed using two very schematic models.very schematic models. A: Water-Oil Displacement in a rectangular A: Water-Oil Displacement in a rectangular

horizontal geometry under gravity forces horizontal geometry under gravity forces dominance (Segregated flow).dominance (Segregated flow).

B: Water-Oil Displacement in a rectangular B: Water-Oil Displacement in a rectangular non- horizontal geometry under gravity forces non- horizontal geometry under gravity forces dominance (Segregated flow).dominance (Segregated flow).


Segregated flow (A). Production at different faces.Segregated flow (A). Production at different faces.

SwirrSwirr

SwirrSwirr

Sw avg = Swirr = 25%Sw avg = Swirr = 25%

QQ

SwSw

QQ

SwSw



SwirrSwirr

SwirrSwirr

Sw avg = 35%Sw avg = 35%

QQ

SwSw

QQ

SwSw



SwirrSwirr

SwirrSwirr


QQ

SwSw

QQ

SwSw


First RemarkFirst Remark

The Relationship between production rate The Relationship between production rate and phases saturation may be different for and phases saturation may be different for different points in the same cell.different points in the same cell.


Segregated flow (B). Production at different faces.Segregated flow (B). Production at different faces.

SwirrSwirr

SwirrSwirr

Sw avg = Swirr = 25%Sw avg = Swirr = 25%

QQ

SwSw

QQ

SwSw



SwirrSwirr

SwirrSwirr


QQ

SwSw

QQ

SwSw


Second RemarkSecond Remark

If capillary forces take part of displacement If capillary forces take part of displacement mechanisms, the relationship between flow mechanisms, the relationship between flow rate and phase saturation may depend on:rate and phase saturation may depend on: The selected production point.The selected production point. The overall geometry and orientation of the The overall geometry and orientation of the

system.system.

nuevos enfoques sobre medición y escalamientos de desplazamientos inmiscibles en medios porosos

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